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Theorem caucvgprlemladdfu 7597
Description: Lemma for caucvgpr 7602. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
caucvgprlemladd.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
caucvgprlemladdfu  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
Distinct variable groups:    A, j    j, F, u, l    n, F, k    k, L, j    S, l, u, j    j,
k
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    S( k, n)    L( u, n, l)

Proof of Theorem caucvgprlemladdfu
Dummy variables  m  r  s  t  v  w  z  f  g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . . . . 7  |-  ( ph  ->  F : N. --> Q. )
2 caucvgpr.cau . . . . . . 7  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
3 caucvgpr.bnd . . . . . . 7  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
4 caucvgpr.lim . . . . . . 7  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
51, 2, 3, 4caucvgprlemcl 7596 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 caucvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 7467 . . . . . . 7  |-  ( S  e.  Q.  ->  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  e.  P. )
86, 7syl 14 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )
9 df-iplp 7388 . . . . . . 7  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
10 addclnq 7295 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvu 7433 . . . . . 6  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
125, 8, 11syl2anc 409 . . . . 5  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
1312biimpa 294 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  E. s  e.  ( 2nd `  L
) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) r  =  ( s  +Q  t ) )
14 breq2 3969 . . . . . . . . . . . . . . . 16  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
1514rexbidv 2458 . . . . . . . . . . . . . . 15  |-  ( u  =  s  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
164fveq2i 5471 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
17 nqex 7283 . . . . . . . . . . . . . . . . . 18  |-  Q.  e.  _V
1817rabex 4108 . . . . . . . . . . . . . . . . 17  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
1917rabex 4108 . . . . . . . . . . . . . . . . 17  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
2018, 19op2nd 6095 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
2116, 20eqtri 2178 . . . . . . . . . . . . . . 15  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
2215, 21elrab2 2871 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2322biimpi 119 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd `  L
)  ->  ( s  e.  Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423adantr 274 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  ->  (
s  e.  Q.  /\  E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2524adantl 275 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( s  e.  Q.  /\ 
E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2625adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  e.  Q.  /\ 
E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2726simpld 111 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
s  e.  Q. )
28 vex 2715 . . . . . . . . . . . . . 14  |-  t  e. 
_V
29 breq2 3969 . . . . . . . . . . . . . 14  |-  ( u  =  t  ->  ( S  <Q  u  <->  S  <Q  t ) )
30 ltnqex 7469 . . . . . . . . . . . . . . 15  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 7470 . . . . . . . . . . . . . . 15  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op2nd 6095 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { u  |  S  <Q  u }
3328, 29, 32elab2 2860 . . . . . . . . . . . . 13  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  S  <Q  t )
34 ltrelnq 7285 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3534brel 4638 . . . . . . . . . . . . 13  |-  ( S 
<Q  t  ->  ( S  e.  Q.  /\  t  e.  Q. ) )
3633, 35sylbi 120 . . . . . . . . . . . 12  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  ( S  e.  Q.  /\  t  e. 
Q. ) )
3736simprd 113 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  e.  Q. )
3837ad2antll 483 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
t  e.  Q. )
3938adantr 274 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  e.  Q. )
40 addclnq 7295 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4127, 39, 40syl2anc 409 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  +Q  t
)  e.  Q. )
42 eleq1 2220 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4342adantl 275 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4441, 43mpbird 166 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  Q. )
4526simprd 113 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )
46 fveq2 5468 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
47 opeq1 3741 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
4847eceq1d 6516 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
4948fveq2d 5472 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
5046, 49oveq12d 5842 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
5150breq1d 3975 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s ) )
5251cbvrexv 2681 . . . . . . . . . 10  |-  ( E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  <->  E. m  e.  N.  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s )
5345, 52sylib 121 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. m  e.  N.  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )
5433biimpi 119 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  S  <Q  t )
5554ad2antll 483 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  ->  S  <Q  t )
5655adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  S  <Q  t )
5756ad2antrr 480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  S  <Q  t )
586ad5antr 488 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  S  e.  Q. )
5939ad2antrr 480 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
t  e.  Q. )
601ad5antr 488 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  F : N. --> Q. )
61 simplr 520 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  m  e.  N. )
6260, 61ffvelrnd 5603 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( F `  m
)  e.  Q. )
63 nnnq 7342 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
64 recclnq 7312 . . . . . . . . . . . . . . . . 17  |-  ( [
<. m ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
6561, 63, 643syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
66 addclnq 7295 . . . . . . . . . . . . . . . 16  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
6762, 65, 66syl2anc 409 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
68 ltanqg 7320 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  Q.  /\  t  e.  Q.  /\  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )  ->  ( S  <Q  t  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t ) ) )
6958, 59, 67, 68syl3anc 1220 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( S  <Q  t  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t ) ) )
7057, 69mpbid 146 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  t ) )
71 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )
72 ltanqg 7320 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
z  <Q  w  <->  ( v  +Q  z )  <Q  (
v  +Q  w ) ) )
7372adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  /\  ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )
)  ->  ( z  <Q  w  <->  ( v  +Q  z )  <Q  (
v  +Q  w ) ) )
7427ad2antrr 480 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
s  e.  Q. )
75 addcomnqg 7301 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  +Q  w
)  =  ( w  +Q  z ) )
7675adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  /\  ( z  e.  Q.  /\  w  e.  Q. )
)  ->  ( z  +Q  w )  =  ( w  +Q  z ) )
7773, 67, 74, 59, 76caovord2d 5990 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q 
( s  +Q  t
) ) )
7871, 77mpbid 146 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q  (
s  +Q  t ) )
79 ltsonq 7318 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
8079, 34sotri 4981 . . . . . . . . . . . . 13  |-  ( ( ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  t )  /\  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q 
( s  +Q  t
) )  ->  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( s  +Q  t
) )
8170, 78, 80syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
s  +Q  t ) )
82 simpllr 524 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
r  =  ( s  +Q  t ) )
8381, 82breqtrrd 3992 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
8483ex 114 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  ->  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  ->  ( (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
8584reximdva 2559 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( E. m  e. 
N.  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  ->  E. m  e.  N.  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q 
r ) )
8653, 85mpd 13 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. m  e.  N.  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
8750oveq1d 5839 . . . . . . . . . 10  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S )  =  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S ) )
8887breq1d 3975 . . . . . . . . 9  |-  ( j  =  m  ->  (
( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
8988cbvrexv 2681 . . . . . . . 8  |-  ( E. j  e.  N.  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r  <->  E. m  e.  N.  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
9086, 89sylibr 133 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
91 breq2 3969 . . . . . . . . 9  |-  ( u  =  r  ->  (
( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u  <->  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
9291rexbidv 2458 . . . . . . . 8  |-  ( u  =  r  ->  ( E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u  <->  E. j  e.  N.  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
9392elrab 2868 . . . . . . 7  |-  ( r  e.  { u  e. 
Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u }  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q 
r ) )
9444, 90, 93sylanbrc 414 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
9594ex 114 . . . . 5  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( r  =  ( s  +Q  t )  ->  r  e.  {
u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9695rexlimdvva 2582 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  ( E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t )  -> 
r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9713, 96mpd 13 . . 3  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
9897ex 114 . 2  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  ->  r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9998ssrdv 3134 1  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   {crab 2439    C_ wss 3102   <.cop 3563   class class class wbr 3965   -->wf 5166   ` cfv 5170  (class class class)co 5824   2ndc2nd 6087   1oc1o 6356   [cec 6478   N.cnpi 7192    <N clti 7195    ~Q ceq 7199   Q.cnq 7200    +Q cplq 7202   *Qcrq 7204    <Q cltq 7205   P.cnp 7211    +P. cpp 7213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-inp 7386  df-iplp 7388
This theorem is referenced by:  caucvgprlemladdrl  7598
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